This 3-in-1 unified thermodynamic and structural framework introduces the Coagulation Rigidity Index (CRI) and Interatomic Spatial Rigidity Index (ISRI) to explain phase transitions, viscosity dynamics, and material rigidity across fluids, solids, and biological systems. Part I: Coagulation as a spatial transition governed by critical temperature T_coag, where interatomic spacing contracts to maximize viscosity and rigidity (CRI ~9.8×10⁷). Applications: blood clotting, protein aggregation, nanofluids, preterm labor diagnostics. Part II: Water’s dual critical behavior – maximum density/viscosity at 4°C (T_coag) and collapse at 96°C (T_collapse). Empirical model μ(T) = 1.55×10⁻³ exp(-((T-32)²/(2·2²))) validated across terrestrial and cryogenic fluids (e.g., Titan subsurface oceans). Part III: ISRI for superhard materials (ReB₂, WB₄, B₄C > diamond hardness) and disordered systems (metallic glasses, graphene foams). FOUNDATIONAL FOR NAVIER-STOKES SMOOTHNESS: CRI viscosity barriers prevent finite-time blow-up in 3D incompressible flows. Companion paper: "Positive and Negative Turbulence in Variable-Geometry Flows" [to be uploaded on Zenodo]. Preprint submitted to Turkish Journal of Physics (NS application). Independent researcher breakthrough toward Clay Millennium Prize. ORCID: 0009-0001-3403-890X